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A003295
McKay-Thompson series of class 11A for the Monster group with a(0) = -5.
(Formerly M3872)
5
1, -5, 17, 46, 116, 252, 533, 1034, 1961, 3540, 6253, 10654, 17897, 29284, 47265, 74868, 117158, 180608, 275562, 415300, 620210, 916860, 1344251, 1953974, 2819664, 4038300, 5746031, 8122072, 11413112, 15943576, 22153909, 30620666
OFFSET
-1,2
COMMENTS
Coefficients of a modular function denoted by B(tau) in Atkin (1967).
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J., 8 (1967), 14-32.
A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J., 8 (1967), 14-32. (Annotated scanned copy, poor quality)
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, see p. 42.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
FORMULA
From Michael Somos, Aug 31 2012: (Start)
Expansion of -11 + (1 + 3*F)^2 * (1/F + 1 + 3*F) where F = eta(q^3) * eta(q^33) / (eta(q) * eta(q^11)) (= g.f. of A128663) in powers of q.
G.f. is Fourier series of a level 11 modular function. f(-1 / (11 t)) = f(t) where q = exp(2 Pi i t).
A000521(n) = a(n) + 11 * a(11*n) unless n=0. [Atkin (1967) p. 22]
a(n) = A003295(n) = A058205(n) = A128525(n) = A134784(n) unless n=0. (End)
a(n) ~ exp(4*Pi*sqrt(n/11)) / (sqrt(2)*11^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 07 2017
EXAMPLE
G.f. = 1/q - 5 + 17*q + 46*q^2 + 116*q^3 + 252*q^4 + 533*q^5 + 1034*q^6 + ...
MATHEMATICA
QP = QPochhammer; F = q*QP[q^3]*(QP[q^33]/(QP[q]*QP[q^11])); s = q*(-11 + (1 + 3*F)^2*(1/F + 1 + 3*F)) + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, from 1st formula *)
PROG
(PARI) q='q+O('q^50); F =q*eta(q^3)*eta(q^33)/(eta(q)*eta(q^11)); Vec(-11 + (1 + 3*F)^2*(3*F + 1 + 1/F)) \\ G. C. Greubel, May 10 2018
KEYWORD
sign,nice,easy
EXTENSIONS
More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 05 2000
STATUS
approved